User gfr - MathOverflow

Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings

2013-05-15T17:00:49Z

If $X$ is a compact oriented surface in a 4-dimensional oriented manifold $M$, then the self-intersection number $X^2$ of $X$ is given by the integral over $X$ of the Euler class of the normal bundle. In the case of $CP^1$ embedded in $CP^2$, the normal bundle is isomorphic to the Hopf bundle, therefore $X^2$ can be obtained calculating the first Chern number of the Hopf fibration (or equivalently the Euler number of its realization).

It is possible to have circle bundles on $CP^1$ with higher Chern number by taking the quotient of the total space of the Hopf fibration by the action generated by $(z^1,z^2)\mapsto(z^1 \exp(i 2 \pi/k), z^2 \exp(i 2\pi/k ))$.

Are these bundles the normal bundles of some embedding of $CP^1$ in a 4-dimensional manifold? If yes, is it possible to describe the embedding explicitly? Is there a deeper relation between the Hopf bundle and the normal bundle of $CP^1$ embedded in $CP^2$ or do they just happen to be the same?

Sign convention in generalised Gauss-Bonnet

2013-03-22T11:35:42Z

Apparently I cannot get the right sign in deriving classical Gauss-Bonnet from generalised one!

According to many references (e.g. Madsen and Tornehave, Nakahara, Milnor and Stasheff if you don't use their particular convention about orientation) generalised Gauss-Bonnet is

$\int_M Pf(-R/2\pi)=\chi(M)$

where $R$ is the curvature 2-form and the Pfaffian of a 2lx2l antisymmetric matrix is $Pf(A)=\frac{1}{2^ll!}\epsilon ^{i_1 \ldots i_{2l}}A_{i_1 i_2} \ldots A_{i_{2l-1}i_{2l}} \ $ i.e. the sign convention is such that for the matrix A={{0,a},{-a,0}} Pf(A)=a.

For l=1, M is a surface and $R^a_{\phantom{a}b}=\frac{1}{2}R^a_{\phantom{a}bcd}e^c\wedge e^d =K (g_{ac}g_{bd}-g_{ad}g_{bc})$ where $K$ is the Gaussian curvature, $g_{ab}$ the metric tensor and ${e^a}$ an orthonormal basis. For the Pfaffian I get $Pf(-R/2 \pi)= -\frac{1}{4\pi} 2 \epsilon_{12} R^1_2= -\frac{1}{2\pi} R^1_{\phantom{1}2}=-\frac{1}{2\pi}R^1_{\phantom{1}212}e^1\wedge e^2$

$e^1\wedge e^2 $ is the volume form and $R^1_{\phantom{1}212}=K$ therefore

$\int_M Pf(-R/2\pi)=\int_M-K/2\pi=\chi$

which obviously has the wrong sign! I have carefully checked the definitions in the books and I think I am using the same as they use - so please be very explicit in pointing aout where the mistake is! Thanks

Understanding four manifolds (more details inside)

2013-03-14T14:57:38Z

I need to understand some of the theory of smooth four manifolds. Eventually I might be interested in learning about Donaldson theory. For the moment I am mainly interested in question such as: if you have a compact, connected, simply-connected four manifold, in how many different ways can I embed a smooth, compact 2-surface in it? A prototype question would be: how many different smooth embeddings of S^2 in the connected sum of CP^2 can I have? I would also like to learn how to calculate self-intersection numbers of 2-surfaces embedded in 4-manifolds.

I have some knowledge of differential geometry and general topology and, at a much lower level, of algebraic topology but my background is in theoretical physics. To give an idea, I feel at home with books like Nakahara "Geometry topology and physics" or "Gravitation" by Misner, Thorne and Wheeler, I like and can follow the more mathematical Naber "Topology, geometry and gauge fields", I find more challenging a book like Bott, Tu "Differential forms in algebraic topology", but I have not worked through it yet (I am more familiar with de Rham theory than anything else in algebraic topology) but e.g. Hatcher "Algebraic Topology" is really though for me.

Given my background, what route would you recommend to learn enough material to allow me to calculate things like those I have pointed out above? I do not want to do research in this area, but be familiar enough with it so that I can use it-see the questions I have raised above for an example of what I mean.

Natural connection on U(1) principal bundles over S^2 with Chern number>1

2012-12-11T12:16:20Z

S^3 can be seen as a U(1)-bundle over S^2 (Hopf fibration). It has first Chern number 1 or -1. Denoting with z_1,z_2 coordinates on C^2 and restricting to S^3 the natural connection form is \omega_1=\bar z_1 dz_1+\bar z_2 dz_2

Consider now the U(1) bundles over S^2 with Chern number n>1. They have total space S^3/Z_n where the Z_n action on C^2 is generated by (z_1,z_2)->(z_1\exp(i2\pi/n),z_2\exp(i2\pi/n)). I know that the appropriate connection form which generalises the one of the Hopf bundle is \omega_n=n(\bar z_1 dz_1+\bar z_2 dz_2) I do not understand where the factor of n comes from. In other terms how do I see, from the defining properties of a connection form (or otherwise), that a factor of n is needed and/or that any other factor would give a one-form which is not a connection form?

Comment by GFR

2013-03-22T15:34:20Z

It turned out that was the problem indeed: my convention for the definition of curvature 2-form where different from those used in the books, resulting in a minus sign difference.

Comment by GFR

2013-03-22T15:27:15Z

I think I have found where the sign comes from: I was using conventions such that the connection form $\omega$ satisfies $de^i+\omega^i_{\phantom{i}j}\wedge e^j=0$, while Milnor-Stasheff has a minus sign, see pag.302. This basically correspond to replace $R^i_{\phantom{i}j} $ with $\R^j_{\phantom{i}i}=-R^i_{\phantom{i}j}$, so that an overall minus results if there is an odd number of $R^i_{\phantom{i}j}$s.

Comment by GFR

2013-03-22T15:21:02Z

That is true but hey also use an unconventional orientation: see page 304, note on signs. Basically if the manifold has dimension 2l and l is even their orientation is the same as the conventional one: $e^1\wedge \ldots \wedge e^{2l}$ but otherwise there is a minus sign. Quoting from the book: "Readers who prefer to use the classical sign conventions as in [Spanier], [Warner] and [Bott-Chern] can forget about these signs, but should replace K with -K whenever it occurs in our characteristic class formulas".

Comment by GFR

2013-03-22T12:40:16Z

Thanks Liviu, I agree that with the conventions that you pointed out everything works out fine. I still have not accepted the answer to see if anyone can confirm that there is a mistake in the books I have mentioned or - more likely - point out what I missed out. In many sources the definition of the Pfaffian is given as $l!A\wedge \ldots \wedge A=Pf(A) vol$ which is equivalent to my definition above.